Varying Degrees of Parallelizability: parallel, sequential

Power Management

Broadcast Scheduling

TCP

Lower Bounds

Braching Programs

Communication Complexity

Cake Cutting

Greedy and Dynamic Programming Algorithms

Embedding Distortion

Time-Space Trade-off Lower Bounds for st-Connectivity on JAG Models

Complexity Classes

Circuit Depth

Comparisons Between Models of Parallel Computation

Randomness

Job DAG

Handling Data

Data Bases

Data Transmission

** Power Management: **
``What matters most to the computer designers at Google is not
speed, but power, low power, because data centers can consume as much
electricity as a city.'' --- Dr. Eric Schmidt, CEO of Google. The
most commonly used power management technique is speed scaling, which
involves changing the speed of each processor. In tandem to
determining at each time, which job to run on each processor, the
operating system will need a speed scaling policy for setting the
speed of each processor.
We consider developing operating system algorithms/policies for
scheduling processes with varying degrees of parallelism in a
multiprocessor setting in a way that both optimizes some schedule
quality of service objective, as well as some power related objective.

** Broadcast Scheduling: **
We investigate server scheduling policies to minimize average user
perceived latency in systems where multiple clients request data from
a server and these are returned on a broadcast channel. Possibly being
able to satisfy many requests to a common file with a single
broadcast, allows the system to be more scalable to large numbers of
clients. One notable commercial example is movies-on-demand. We
provide new on-line scheduling algorithms and new analysis techniques.

** TCP: **
The standard for reliable communication on the Internet is the
well-known Transport Control Protocol (TCP). It performs well both in
practice and extensive simulation have been done, but there have been
minimal theoretical results. We evaluate the performance of TCP using
the traditional competitive analysis framework, comparing its
performance with that of an optimal offline algorithm.

* Warning:
I do not do TCP AT ALL.
I am a theory person. I prove theorems.

** Braching Programs:**
The most powerful model of computation measuring the
amount of space used by an algorithm is branching programs. It is
represented by a directed acyclic graph of states. At each point in
time, the computation is in a particular state. Imagine that the name
of this state specifies everything the computation knows. For example,
being in state q

** Communication Complexity:**
Suppose Alice knows some information x and Bob knows y and together
they want to solve some task f(x,y). Communication complexity measures
the number of bits they must communicate in order to achieve this.
Shannon's entropy is a good measure of the amount of information sent.
We prove a number of results in this area. One of them looks at how much a
little advice can help Alice and Bob.

** Cake Cutting:**
The fair division of resources is an important topic, be it settling a
divorce, resolving international issues such as contested underwater
mining territories, or simply cutting a cake. The task is about
figuring out how to divide the resource so that each of the n
recipient feels they've received a fair portion based on their needs
and desires. The best algorithm achieves this with O(n log n)
operations, where each operation either asks a recipient to either
specify how much they value a particular piece or where they would cut
the cake to produce a piece of a given value. We prove a matching
lower bound. We also reduced the time to only O(n) operations by
allowing the algorithm to flip coins and to provide an approximately
fair solution. Interestingly enough, this work was written about in
the Toronto Star [Feb 12, 2006. pg. D.16]
We give a faster randomized algorithm that is in
https://en.wikipedia.org/wiki/Edmonds-Pruhs_protocol

** Greedy and Dynamic Programming Algorithms: **
The goal of this research area is to define a model of computation,
prove that it to some extent captures an algorithm paradigm (in this
case greedy algorithms or dynamic programming) because many of the key
algorithms in this paradigm can be implemented in the model, and then
to prove that other computational problems cant be solved efficiently
in this model. We have a number of results in this area.

** Embedding Distortion: **
The adversary places n points in the plane. For every pair of points,
he tells me the distance between each pair distorted by factor of at
most 1+epsilon. My task, without knowing the original placement, is to
again place the points in plane in a way that respects the stated
distances within a factor of 1+epsilon'. Surely for small epsilon and
reasonably large epsilon', the problem is easy. Surprisingly, we did
prove that the problem is NP-hard when 1+epsilon is 3.62 and
1+epsilon' is 3.99. My conjecture, is that it is poly-time for
epsilon' > 2 epsilon and NP-hard for epsilon' < 2 epsilon.

** Time-Space Trade-off Lower Bounds for st-Connectivity on JAG
Models: **
The computational problem st-connectivity is to determine whether
there is a path from vertex s to vertex t in a graph. The model
considered is the JAG (``jumping automaton for graphs'') introduced by
Cook and Rackoff in which pebbles are moved around the vertices of the
input graph. It is a very natural structured model and is powerful
enough so that most known algorithms can be implemented on it. We
prove tight lower bounds on how the required time increases
exponentially as the space (# of memory cells) available decreases.

** Complexity Classes: **
Instead of studying classes of decision problems based on their
computation times, Papadimitriou, Schafer, and Yannakakis categorized
search problems into a number of complexity classes. This is
particularly important when the associated decision problem is not
known to be in P nor to be NP-complete and when the object being
searched for is known to always exist. Our paper studies these
classes and proves some equivalences and separations among them.

** Circuit Depth: **
Parallel computation is important but not well understood. The next
significant hurdle is proving that there is a problem that cannot be
solved on a circuit (and, or, not) with only O(log n) depth
(i.e. separate NC^1 from NC^2). Towards this goal, Karchmer, Raz, and
Wigderson suggested the intermediate step of proving a lower bound on
a slightly simplified version of their communication game
characterization of circuit depth, which they call the Universal
Composition Relation. Our paper gives an almost optimal lower bound
for this intermediate problem.

** Comparisons Between Models of Parallel Computation: **
For my masters, I used Ramsey theory and information theory to
separate to models of parallel computation. This provided a greater
understanding of the partial information a processor learns about the
input.

** Job DAG: **
We consider the problem of computing bounds on the variance and
expectation of the longest path length in a DAG from knowledge of
variance and expectation of edge lengths. We present analytic bounds
for various simple DAG structures, and present a new algorithm to
compute bounds for more general DAG structures. Our algorithm is
motivated by an analogy with balance of forces in a network of
``strange'' springs. The problem has applications in reliability
analysis and probabilistic verification of interface systems.

** Data Bases: ** As part of data mining, we give a fast single
scan algorithm for finding all maximal empty rectangles in large
two-dimensional data sets. IBM bought the patent from us.

** Data Transmission: **
In current transmission networks, packets of data often get lost in
chaotic bursts. Applications such as video conferencing and video on
demand suffer because the image ``can break up with a jarring
abruptness'' even when a only few packets have been lost. We introduce
a novel approach called Priority Encoding Transmission (PET) that
ensures the video image will degrade gracefully with the number of
packets lost, independent of which packets are lost.